On the Cauchy problem to the axially-symmetric solutions to the Navier-Stokes equations

TL;DR

This paper establishes the existence of global regular solutions to the axisymmetric Navier-Stokes equations by deriving a priori estimates near and far from the axis of symmetry, using expansions and energy estimates.

Contribution

It provides new global a priori estimates for axisymmetric Navier-Stokes solutions, advancing understanding of regularity near the symmetry axis.

Findings

Global a priori estimate for vorticity components

Existence of global regular solutions under certain regularity conditions

Derivation of higher regularity estimates for velocity and pressure

Abstract

We consider the Cauchy problem to the axisymmetric Navier-Stokes equations. To prove an existence of global regular solutions we examine the Navier-Stokes equations near the axis of symmetry and far from it separately. We derive only a global a priori estimate. To show it near the axis of symmetry we need the energy estimate, $L_\infty$-estimate for swirl, $H^2$ and $H^3$ estimates for the modified stream function (stream function divided by radius) and also expansions of velocity and modified stream function found by Liu-Wang. The estimate for solutions far from the axis of symmetry follows easily. Hence, having so regular solutions that Liu-Wang expansions hold we have the global a priori estimate $(\Omega=\mathbb{R}^3)$ $$ \|\omega_{r/r} \|_{V(\Omega^t)} + \|\omega_{\varphi/r}\|_{V(\Omega^t)}\le\phi({\rm data}),\ \ t<\infty, \qquad(*) $$ where $\omega_r$ is the radiar component of vorticity, $\omega_\varphi$ the angular, $V(\Omega^t)$ is the energy norm. Estimate $(*)$ can be treated as an a priori estimate derived on sufficiently regular solutions. Increasing regularity of solutions $(*)$ we derive the estimate $$\eqalign{ &\|v\|_{W_3^{3,3/2}(\Omega^t)}+\|\nabla p\|_{W_3^{1,1/2}(\Omega^t)}\cr &\le\phi(\phi({\rm data}),\|f\|_{W_3^{1,1/2}(\Omega^t)},\|v(0)\|_{W_3^{3-2/3}(\Omega)}),\cr} \qquad(**) $$ where $\phi$ is an increasing positive function. The estimate is proved on the local solution. Estimate $(**)$ plus existence of local solutions imply the existence of global regular solutions to the Cauchy problem.